Sigma-Prikry forcing I: The Axioms

Joint work with Alejandro Poveda and Dima Sinapova.

Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that given a $\Sigma$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $\Sigma$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma$-Prikry posets. Putting the two works together, we obtain a proof of the following.

Theorem. If $\kappa$ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa$ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa^+$ reflects simultaneously, and $2^\kappa=\kappa^{++}$.

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A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math., 73(5): 1205-1238, 2021.